2 Answers
2

Pretty much coaster brakes themselves are interchangeable. There is an arm that comes out of the hub and is attached to the chain stay to counteract the torque of the brake. Generally the attachment is made with a strap, so a little difference in arm length is not a problem. (If you have the rare frame where the arm is attached to a tab on the frame, though, you may want to check arm length.)

The important things are then:

The sprocket size. Pretty much all coaster brake chains are the same width/pitch, but the number of teeth on the cog can differ.

The axle diameter. I think this is pretty standard, but likely there's a slight difference between English and metric dimensioned axles.

The axle width, measured between the outside edge of the nuts that fit inside the frame. Doesn't need to be exact, but within 1/4" or so. Again, this should be fairly standard.

The rim diameter/width. As the other answers note, there's a significant difference between a 2-1/4" rim and a 2.125" rim -- two different systems for sizing rims, and it affects rim diameter more than rim width.

Note that if you can find just the rim you can get the existing wheel rebuilt.

Because you're using a coaster brake, rim diameter is less critical an issue than in most cases. The size 24 x 2.125 is actually a tire size which corresponds to 54-507 in ISO measurements. That means that the bead seat diameter (diameter of the rim) is 507mm and the tire is 54mm wide. If the original wheel is ~24", it will correspond to BSD sizes between 520mm - 547mm (because there are lots of "24" size tires, read Sheldon Brown.) The new rim will be smaller...however... with a wider tire, it may make for a close fit. Match the old tire size with the link and you'll have a better idea of the original wheel-tire size as compared with your potential new size. Keep in mind that you can buy rather narrow tires to provide clearance for your fenders.

Another consideration is the Over Locknut Dimension (O.L.D. for short) which is how wide the hub is at the axle nuts (and correspondingly, how wide the frame is.) This is something that you'll need to break out a decent metric ruler to determine. You want to get that dimension close, but if it is off it isn't a big deal. +/- 5mm is an acceptable stretch. Anything more and you may want to consider either finding the correct width hub or swaging out the seat/chain stays to accept a wider or smaller hub (steel is a very forgiving frame metal.) Another option is to use a longer axle and spacers in a hub that is too narrow...something that you may want a shop to handle if you've never rebuilt a hub before, especially considering it will alter the chain-line.